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Chapter 2: Problem 35

Solve each equation, and check the solution. \(11 x-5(x+2)=6 x+5\)

Short Answer

Expert verified
No solution.

Step by step solution

01

Distribute in the equation

Distribute the \text{-5} across the \text{(x + 2)} term on the left side of the equation. This gives us: \[ 11x - 5(x + 2) = 6x + 5 \] Applying the distribution, \[ 11x - 5x - 10 = 6x + 5 \]
02

Combine like terms

Combine the like terms on the left side of the equation. This becomes: \[ (11x - 5x) - 10 = 6x + 5 \] Which simplifies to: \[ 6x - 10 = 6x + 5 \]
03

Eliminate the variable term

Subtract \text{6x} from both sides of the equation to eliminate the variable term from one side: \[ 6x - 10 - 6x = 6x + 5 - 6x \] This simplifies to: \[ -10 = 5 \]
04

Analyze the result

The resulting statement \[ -10 = 5 \] is false. Therefore, there is no solution to this equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

No Solution
In algebra, you may come across an equation where, after working through all the usual steps, you end up with a statement that is clearly false. This means that the equation has no solution. In this exercise, after simplifying the equation, we get 鈭�10 = 5. Since this statement is impossible (because 鈭�10 is never equal to 5), we conclude that there is no value of x that will satisfy the original equation. **Having no solution** can occur when the variables cancel out and leave behind a false numerical statement.
It's important to recognize when this happens to avoid wasting time searching for a solution that does not exist.
Distribution in Algebra
The distributive property is a key concept in algebra that allows you to simplify expressions and equations. It states that a(b + c) = ab + ac can be distributed to both b and c. In our example, we used distribution to expand 鈥� 5(x + 2).
Here's how:
  • Start with the expression 鈥�5(x + 2)
  • Multiply 鈥�5 by x, resulting in 鈥�5x
  • Multiply 鈥�5 by 2, resulting in 鈥�10
    Combine these to get 鈥�5x鈥�10
This step makes it easier to combine like terms and simplify the equation later on.
Remember, distribution is useful for breaking down complex expressions and should be applied carefully to ensure accurate results.
Combining Like Terms
In algebra, **combining like terms** is a process used to simplify expressions and make solving equations easier. Like terms are terms that have the same variable raised to the same power. For instance, 6x and 鈥�3x are like terms because they both have the variable x. In our example, after distributing, we combined like terms on the left side of the equation:

Starting with: 11x 鈥� 5x 鈥� 10 = 6x + 5
Combine 11x and 鈥�5x to get 6x:
  • This simplifies the left side to 6x 鈥� 10
Combining like terms simplifies the equation and helps isolate the variable, making it easier to solve.
Being proficient in this skill can make algebra problems much more manageable.

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